This is a problem with your approach via completing squares. These are integers, you cannot start treating them as rational numbers.

I understand that. I was being lazy with my notation. Obviously when I multiply a number by I am multiplying it by 17, the multiplicative inverse of 9 in modulo 19. (I can get away with this only because 19 is prime, thus is a field.)

However it doesn't matter much since my method didn't answer any of his questions. I just posted it because I didn't realize your post addressed the question (I didn't realize that until later, I thought it was in reference to his last question), and wanted to tell him the answer is that there is no solution to the equation so that he would know what to expect as an answer when he used the methods he was required to use.